Paper 2020/1605

$P_4$-free Partition and Cover Numbers and Application

Alexander R. Block, Simina Branzei, Hemanta K. Maji, Himanshi Mehta, Tamalika Mukherjee, and Hai H. Nguyen

Abstract

$P_4$-free graphs-- also known as cographs, complement-reducible graphs, or hereditary Dacey graphs--have been well studied in graph theory. Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite $P_4$-free graphs. For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant. Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems. Interestingly, our covering problem is closely related to the well-studied problem of product/Prague dimension of loopless undirected graphs, which allows us to employ algebraic lower-bounding techniques for the product/Prague dimension. We prove that computing these numbers is \npol-complete, even for bipartite graphs. We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size-$N$ partite sets such that these numbers are at least ${\epsilon\cdot N^{1-2\epsilon}}$, for $\epsilon\in\{1/3,1/4,1/5,\dotsc\}$. Finally, we accurately estimate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality. For applications in information theory and communication \& cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the participants, Alice and Bob, their portion from this joint sample. Alice and Bob's objective is to non-interactively establish a shared key and extract the left-over entropy from their portion of the samples as independent private randomness. A genie, who observes the joint sample, provides appropriate assistance to help Alice and Bob with their objective. Lower bounds to the minimum size of the genie's assistance translate into communication and cryptographic lower bounds. We show that (the $\log_2$ of) the $P_4$-free partition number of a graph encoding the joint distribution that the setup uses is equivalent to the size of the genie's assistance. Consequently, the joint distributions corresponding to the bipartite graphs constructed above with high $P_4$-free partition numbers correspond to joint distributions requiring more assistance from the genie. As a representative application in non-deterministic communication complexity, we study the communication complexity of nondeterministic protocols augmented by access to the equality oracle at the output. We show that (the $\log_2$ of) the $P_4$-free cover number of the bipartite graph encoding a Boolean function $f$ is equivalent to the minimum size of the nondeterministic input required by the parties (referred to as the communication complexity of $f$ in this model). Consequently, the functions corresponding to the bipartite graphs with high $P_4$-free cover numbers have high communication complexity. Furthermore, there are functions with communication complexity close to the \naive protocol where the nondeterministic input reveals a party's input. Finally, the access to the equality oracle reduces the communication complexity of computing set disjointness by a constant factor in contrast to the model where parties do not have access to the equality oracle. To compute the inequality function, we show an exponential reduction in the communication complexity, and this bound is optimal. On the other hand, access to the equality oracle is (nearly) useless for computing set intersection.